Haldane's formula in Cannings models: The case of moderately weak selection

Abstract

We introduce a Cannings model with directional selection via a paintbox construction and establish a strong duality with the line counting process of a new \emph{Cannings ancestral selection graph} in discrete time. This duality also yields a formula for the fixation probability of the beneficial type. Haldane's formula states that for a single selectively advantageous individual in a population of haploid individuals of size $N$ the prob\-ability of fixation is asymptotically (as $N\to \infty$) equal to the selective advantage of haploids $s_N$ divided by half of the offspring variance. For a class of offspring distributions within Kingman attraction we prove this asymptotics for sequences $s_N$ obeying $N^{-1}\ll s_N\ll N^{-1/2}$, which is a regime of "moderately weak selection". It turns out that for $s_N\ll N^{-2/3}$ the Cannings ancestral selection graph is so close to the ancestral selection graph of a Moran model that a suitable coupling argument allows to play the problem back asymptotically to the fixation probability in the Moran model, which can be computed explicitly.

Minor revision of the former version. In particular, we made the following changes: Condition (3.8) in this version is slightly weaker than Condition (3.9) in the former version. The proof of Theorem 3.5b remained essentially the same. The former Section 4 is now part of Section 2 (Section 2.4). Lemma 6.3 and 6.4 are interchanged, now Lemma 5.4 and 5.3